Introduction to Convex Analysis

This chapter covers the elementary theory of convex sets in ℝ n $$\mathbb {R}^{n}$$ , together with a quite detailed introduction to convex and concave functions. Section 3.3 is concerned with two generalizations of the class of convex functions, i.e. the class of quasiconvex functions and the class of pseudoconvex functions. Quasiconvex (and quasiconcave) functions are important in optimization theory, but perhaps are more important in economic theory, namely utility theory and other topics. Pseudoconvex functions are important in optimization problems, as this class of generalized convex functions mantains the main properties of convex functions, with respect to optimal points. Section 3.4 briefly presents the main separation theorems between convex sets and the main theorems of the alternative for linear systems, such as the basic Farkas-Minkowski theorem, an important tool in optimization theory. Section 3.5 presents the notion of extremal points of a convex set, useful in the theory of Linear Programming (Chap. 5 ).